Worksheet: Linear Transformation Composition

Download

In this worksheet, we will practice finding the matrix of two or more consecutive linear transformations.

Q1:

Find the matrix with respect to the standard basis vectors for the linear transformation that rotates every vector in ℝ through an angle of 𝜋3 and then reflects it across the 𝑦-axis.

  • A ⎡ ⎢ ⎢ ⎢ ⎣ − 1 2 √ 3 2 √ 3 2 1 2 ⎤ ⎥ ⎥ ⎥ ⎦
  • B ⎡ ⎢ ⎢ ⎢ ⎣ 1 2 √ 3 2 √ 3 2 1 2 ⎤ ⎥ ⎥ ⎥ ⎦
  • C ⎡ ⎢ ⎢ ⎢ ⎣ 1 2 − √ 3 2 √ 3 2 1 2 ⎤ ⎥ ⎥ ⎥ ⎦
  • D ⎡ ⎢ ⎢ ⎢ ⎣ − √ 3 2 1 2 1 2 √ 3 2 ⎤ ⎥ ⎥ ⎥ ⎦
  • E ⎡ ⎢ ⎢ ⎢ ⎣ − √ 3 2 1 2 √ 3 2 1 2 ⎤ ⎥ ⎥ ⎥ ⎦

Q2:

A linear transformation is formed by reflecting every vector in ℝ in the 𝑦-axis and then rotating the resulting vector through an angle of 𝜋4. Find the matrix of this linear transformation.

  • A ⎡ ⎢ ⎢ ⎢ ⎣ − √ 2 2 √ 2 2 − √ 2 2 √ 2 2 ⎤ ⎥ ⎥ ⎥ ⎦
  • B ⎡ ⎢ ⎢ ⎢ ⎣ − √ 2 2 0 0 √ 2 2 ⎤ ⎥ ⎥ ⎥ ⎦
  • C ⎡ ⎢ ⎢ ⎢ ⎣ − √ 2 2 √ 2 2 √ 2 2 √ 2 2 ⎤ ⎥ ⎥ ⎥ ⎦
  • D ⎡ ⎢ ⎢ ⎢ ⎣ − √ 2 2 − √ 2 2 − √ 2 2 √ 2 2 ⎤ ⎥ ⎥ ⎥ ⎦
  • E ⎡ ⎢ ⎢ ⎢ ⎣ √ 2 2 − √ 2 2 √ 2 2 √ 2 2 ⎤ ⎥ ⎥ ⎥ ⎦

Q3:

A linear transformation is formed by rotating every vector in ℝ through an angle of 𝜋6, reflecting the resulting vector in the 𝑥-axis, and finally reflecting this vector in the 𝑦-axis. Find the matrix of this linear transformation.

  • A ⎡ ⎢ ⎢ ⎢ ⎣ − √ 3 2 0 0 √ 3 2 ⎤ ⎥ ⎥ ⎥ ⎦
  • B ⎡ ⎢ ⎢ ⎢ ⎣ − √ 3 2 − 1 2 1 2 √ 3 2 ⎤ ⎥ ⎥ ⎥ ⎦
  • C ⎡ ⎢ ⎢ ⎢ ⎣ √ 3 2 − 1 2 1 2 √ 3 2 ⎤ ⎥ ⎥ ⎥ ⎦
  • D ⎡ ⎢ ⎢ ⎢ ⎣ − √ 3 2 1 2 − 1 2 − √ 3 2 ⎤ ⎥ ⎥ ⎥ ⎦
  • E ⎡ ⎢ ⎢ ⎢ ⎣ √ 3 2 − 1 2 − 1 2 − √ 3 2 ⎤ ⎥ ⎥ ⎥ ⎦

Q4:

A linear transformation is formed by rotating every vector in ℝ through an angle of 2𝜋3 and then reflecting the resulting vector in the 𝑥-axis. Find the matrix of this linear transformation.

  • A ⎡ ⎢ ⎢ ⎢ ⎣ − 1 2 − √ 3 2 √ 3 2 − 1 2 ⎤ ⎥ ⎥ ⎥ ⎦
  • B ⎡ ⎢ ⎢ ⎣ − 1 2 0 0 1 2 ⎤ ⎥ ⎥ ⎦
  • C ⎡ ⎢ ⎢ ⎢ ⎣ 1 2 − √ 3 2 − √ 3 2 − 1 2 ⎤ ⎥ ⎥ ⎥ ⎦
  • D ⎡ ⎢ ⎢ ⎢ ⎣ − 1 2 √ 3 2 − √ 3 2 1 2 ⎤ ⎥ ⎥ ⎥ ⎦
  • E ⎡ ⎢ ⎢ ⎢ ⎣ − 1 2 − √ 3 2 − √ 3 2 1 2 ⎤ ⎥ ⎥ ⎥ ⎦

Q5:

A linear transformation is formed by reflecting every vector in ℝ across the 𝑦-axis and then rotating the resulting vector through an angle of 𝜋6. Find the matrix of this linear transformation.

  • A ⎡ ⎢ ⎢ ⎢ ⎣ − √ 3 2 0 0 √ 3 2 ⎤ ⎥ ⎥ ⎥ ⎦
  • B ⎡ ⎢ ⎢ ⎢ ⎣ − √ 3 2 − 1 2 − 1 2 √ 3 2 ⎤ ⎥ ⎥ ⎥ ⎦
  • C ⎡ ⎢ ⎢ ⎢ ⎣ √ 3 2 − 1 2 1 2 √ 3 2 ⎤ ⎥ ⎥ ⎥ ⎦
  • D ⎡ ⎢ ⎢ ⎢ ⎣ − √ 3 2 1 2 1 2 √ 3 2 ⎤ ⎥ ⎥ ⎥ ⎦
  • E ⎡ ⎢ ⎢ ⎢ ⎣ − √ 3 2 1 2 − 1 2 √ 3 2 ⎤ ⎥ ⎥ ⎥ ⎦

Q6:

A linear transformation is formed by rotating every vector in ℝ through an angle of 𝜋3 and then reflecting the resulting vector in the 𝑥-axis. Find the matrix of this linear transformation.

  • A ⎡ ⎢ ⎢ ⎣ 1 2 0 0 − 1 2 ⎤ ⎥ ⎥ ⎦
  • B ⎡ ⎢ ⎢ ⎢ ⎣ 1 2 − √ 3 2 √ 3 2 1 2 ⎤ ⎥ ⎥ ⎥ ⎦
  • C ⎡ ⎢ ⎢ ⎢ ⎣ − 1 2 − √ 3 2 − √ 3 2 1 2 ⎤ ⎥ ⎥ ⎥ ⎦
  • D ⎡ ⎢ ⎢ ⎢ ⎣ 1 2 − √ 3 2 − √ 3 2 − 1 2 ⎤ ⎥ ⎥ ⎥ ⎦
  • E ⎡ ⎢ ⎢ ⎢ ⎣ 1 2 √ 3 2 − √ 3 2 − 1 2 ⎤ ⎥ ⎥ ⎥ ⎦

Q7:

A linear transformation is formed by reflecting every vector in ℝ in the 𝑥-axis and then rotating the resulting vector through an angle of 𝜋6. Find the matrix of this linear transformation.

  • A ⎡ ⎢ ⎢ ⎢ ⎣ √ 3 2 0 0 − √ 3 2 ⎤ ⎥ ⎥ ⎥ ⎦
  • B ⎡ ⎢ ⎢ ⎢ ⎣ √ 3 2 1 2 1 2 − √ 3 2 ⎤ ⎥ ⎥ ⎥ ⎦
  • C ⎡ ⎢ ⎢ ⎢ ⎣ √ 3 2 − 1 2 1 2 − √ 3 2 ⎤ ⎥ ⎥ ⎥ ⎦
  • D ⎡ ⎢ ⎢ ⎢ ⎣ √ 3 2 − 1 2 − 1 2 − √ 3 2 ⎤ ⎥ ⎥ ⎥ ⎦
  • E ⎡ ⎢ ⎢ ⎢ ⎣ √ 3 2 − 1 2 1 2 √ 3 2 ⎤ ⎥ ⎥ ⎥ ⎦

Q8:

A linear transformation is formed by reflecting every vector in ℝ in the 𝑥-axis and then rotating the resulting vector through an angle of 𝜋4. Find the matrix of this linear transformation.

  • A ⎡ ⎢ ⎢ ⎢ ⎣ √ 2 2 − √ 2 2 − √ 2 2 − √ 2 2 ⎤ ⎥ ⎥ ⎥ ⎦
  • B ⎡ ⎢ ⎢ ⎢ ⎣ √ 2 2 0 0 − √ 2 2 ⎤ ⎥ ⎥ ⎥ ⎦
  • C ⎡ ⎢ ⎢ ⎢ ⎣ √ 2 2 √ 2 2 √ 2 2 − √ 2 2 ⎤ ⎥ ⎥ ⎥ ⎦
  • D ⎡ ⎢ ⎢ ⎢ ⎣ √ 2 2 − √ 2 2 √ 2 2 − √ 2 2 ⎤ ⎥ ⎥ ⎥ ⎦
  • E ⎡ ⎢ ⎢ ⎢ ⎣ √ 2 2 − √ 2 2 √ 2 2 √ 2 2 ⎤ ⎥ ⎥ ⎥ ⎦

Q9:

Let the matrix 𝐴 represent rotation in the plane through an angle of 𝜃 and let the matrix 𝐵 represent reflection in the 𝑥-axis.

What is the matrix 𝐴?

  • A 𝜃 𝜃 − 𝜃 𝜃 c o s s i n s i n c o s
  • B 𝜃 𝜃 𝜃 𝜃 c o s s i n s i n c o s
  • C 𝜃 − 𝜃 𝜃 𝜃 s i n c o s c o s s i n
  • D 𝜃 − 𝜃 𝜃 𝜃 c o s s i n s i n c o s
  • E 𝜃 𝜃 𝜃 𝜃 s i n c o s c o s s i n

What is the matrix 𝐵?

  • A − 1 0 0 − 1
  • B − 1 0 0 1
  • C 0 1 1 0
  • D 0 − 1 − 1 0
  • E 1 0 0 − 1

What is the matrix 𝐴𝐵?

  • A 𝜃 𝜃 𝜃 − 𝜃 c o s s i n s i n c o s
  • B 𝜃 − 𝜃 𝜃 𝜃 c o s s i n s i n c o s
  • C 𝜃 − 𝜃 𝜃 𝜃 s i n c o s c o s s i n
  • D 𝜃 𝜃 − 𝜃 𝜃 c o s s i n s i n c o s
  • E 𝜃 𝜃 𝜃 − 𝜃 s i n c o s c o s s i n

Q10:

Suppose that the matrix 𝐴 represents rotation about the origin through an angle of 𝜃 (measuring between 0∘ and 90∘) and the matrix 𝐵 represents reflection in the 𝑥-axis.

Find 𝑀=𝐴𝐵.

  • A 𝑀 = 𝐴 𝐵 = − 𝜃 𝜃 − 𝜃 𝜃 c o s s i n s i n c o s
  • B 𝑀 = 𝐴 𝐵 = − 𝜃 − 𝜃 𝜃 − 𝜃 c o s s i n s i n c o s
  • C 𝑀 = 𝐴 𝐵 = 𝜃 𝜃 𝜃 − 𝜃 c o s s i n s i n c o s
  • D 𝑀 = 𝐴 𝐵 = 𝜃 − 𝜃 𝜃 − 𝜃 c o s s i n s i n c o s
  • E 𝑀 = 𝐴 𝐵 = 𝜃 − 𝜃 𝜃 𝜃 c o s s i n s i n c o s

Note that 𝑀 is a reflection in a line through the origin. Let this line of reflection have equation 𝑦=𝑘𝑥. By considering the image of the vector ⟨1,0⟩, determine the measure of the angle between the 𝑥-axis and the line of reflection.

  • A 𝜋 2 − 𝜃
  • B 𝜃 2
  • C − 𝜃 2
  • D 𝜃
  • E − 𝜃

What, therefore, is the slope 𝑘 of the line of reflection?

  • A t a n 𝜃
  • B t a n 𝜃 2
  • C − 𝜃 t a n
  • D − 𝜃 2 t a n
  • E c o t 𝜃

If v lies in the direction of the line of reflection, what is 𝑀v?

  • A ⟨ 1 , 0 ⟩
  • B − v
  • C ⟨ 0 , 1 ⟩
  • D v
  • E ⟨ 0 , 0 ⟩

By solving the equation obtained in the previous part, find another expression for the slope 𝑘 of the line of reflection.

  • A 𝑘 = 𝜃 1 + 𝜃 = 1 − 𝜃 𝜃 s i n c o s c o s s i n
  • B 𝑘 = 𝜃 1 − 𝜃 = 1 + 𝜃 𝜃 c o s s i n s i n c o s
  • C 𝑘 = 𝜃 1 − 𝜃 = 1 + 𝜃 𝜃 s i n c o s c o s s i n
  • D 𝑘 = 𝜃 1 − 𝜃 = 1 − 𝜃 𝜃 s i n c o s c o s s i n
  • E 𝑘 = 𝜃 1 + 𝜃 = 1 − 𝜃 𝜃 c o s s i n c o s s i n

Q11:

A linear transformation is formed by rotating every vector in ℝ through an angle of 30∘ counterclockwise (when viewed from the positive 𝑧-axis) about the 𝑧-axis and then reflecting the resulting vector in the 𝑥𝑦-plane. Find the matrix of this linear transformation.

  • A ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ √ 3 2 − 1 2 0 1 2 √ 3 2 0 0 0 − 1 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
  • B ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ √ 3 2 − 1 2 0 1 2 √ 3 2 0 0 0 1 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
  • C ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 1 2 − √ 3 2 0 √ 3 2 1 2 0 0 0 1 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
  • D ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 1 2 − √ 3 2 0 √ 3 2 1 2 0 0 0 − 1 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
  • E ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ √ 3 2 − 1 2 0 − 1 2 − √ 3 2 0 0 0 − 1 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

Q12:

A linear transformation is formed by rotating every vector in ℝ through an angle of 𝜋4 and then reflecting the resulting vector in the 𝑥-axis. Find the matrix of this linear transformation.

  • A ⎡ ⎢ ⎢ ⎢ ⎣ √ 2 2 0 0 − √ 2 2 ⎤ ⎥ ⎥ ⎥ ⎦
  • B ⎡ ⎢ ⎢ ⎢ ⎣ − √ 2 2 √ 2 2 √ 2 2 √ 2 2 ⎤ ⎥ ⎥ ⎥ ⎦
  • C ⎡ ⎢ ⎢ ⎢ ⎣ √ 2 2 √ 2 2 √ 2 2 √ 2 2 ⎤ ⎥ ⎥ ⎥ ⎦
  • D ⎡ ⎢ ⎢ ⎢ ⎣ √ 2 2 − √ 2 2 √ 2 2 √ 2 2 ⎤ ⎥ ⎥ ⎥ ⎦
  • E ⎡ ⎢ ⎢ ⎢ ⎣ √ 2 2 − √ 2 2 − √ 2 2 − √ 2 2 ⎤ ⎥ ⎥ ⎥ ⎦

Q13:

A vector in ℝ is rotated counterclockwise about the origin through an angle of 2𝜋3, and the result is reflected in the 𝑥-axis. Find, with respect to the standard basis, the matrix of this combined transformation.

  • A ⎡ ⎢ ⎢ ⎢ ⎣ − √ 3 2 1 2 √ 3 2 1 2 ⎤ ⎥ ⎥ ⎥ ⎦
  • B ⎡ ⎢ ⎢ ⎢ ⎣ − 1 2 − √ 3 2 − √ 3 2 1 2 ⎤ ⎥ ⎥ ⎥ ⎦
  • C ⎡ ⎢ ⎢ ⎢ ⎣ 1 2 − √ 3 2 √ 3 2 − 1 2 ⎤ ⎥ ⎥ ⎥ ⎦
  • D ⎡ ⎢ ⎢ ⎢ ⎣ 1 2 √ 3 2 √ 3 2 1 2 ⎤ ⎥ ⎥ ⎥ ⎦
  • E ⎡ ⎢ ⎢ ⎢ ⎣ √ 3 2 − 1 2 1 2 − √ 3 2 ⎤ ⎥ ⎥ ⎥ ⎦

Q14:

Suppose 𝐴 and 𝐵 are 2×2 matrices, with 𝐴 representing a counterclockwise rotation of 30∘ about the origin and 𝐵 representing a reflection in the 𝑥-axis. What does the matrix 𝐵𝐴 represent?

  • Aa reflection in the line through the origin at a 75∘ inclination
  • Ba reflection in the line through the origin at a −15∘ inclination
  • Ca reflection in the line through the origin at a −75∘ inclination
  • Da reflection in the line through the origin at a −45∘ inclination
  • Ea reflection in the line through the origin at a 15∘ inclination

Q15:

Suppose 𝐴 and 𝐵 are 2×2 matrices, with 𝐴 representing a counterclockwise rotation of 30∘ about the origin and 𝐵 representing a reflection in the 𝑥-axis. What does the matrix 𝐴𝐵 represent?

  • Aa reflection in the line through the origin at a −15∘ inclination
  • Ba reflection in the line through the origin at a −75∘ inclination
  • Ca reflection in the line through the origin at a 15∘ inclination
  • Da reflection in the line through the origin at a 75∘ inclination
  • Ea reflection in the line through the origin at a −45∘ inclination

Q16:

Describe the geometric effect of the transformation produced by the matrix 0−330.

  • Aa dilation with center the origin and scale factor 3 followed by a reflection in the line 𝑦=−𝑥
  • Ba dilation with center the origin and scale factor 3 followed by a 180∘ rotation about the origin
  • Ca dilation with center the origin and scale factor 3 followed by a reflection in the line 𝑦=𝑥
  • Da dilation with center the origin and scale factor 3 followed by a 90∘ rotation about the origin
  • Ea dilation with center the origin and scale factor 3 followed by a −90∘ rotation about the origin

Q17:

Which of the following compositions of transformations is represented by the matrix 0−2−20?

  • Aa dilation with center the origin and scale factor 2 followed by a reflection in the line 𝑦=−𝑥
  • Ba rotation by 180∘ about the origin followed by a reflection in the line 𝑦=𝑥
  • Ca dilation with center the origin and scale factor 2 followed by a reflection in the line 𝑦=𝑥
  • Da dilation with center the origin and scale factor −2 followed by a reflection in the line 𝑦=−𝑥
  • Ea dilation with center the origin and scale factor −2 followed by a reflection in the line 𝑦=𝑥

Q18:

A dilation with center the origin is composed with a rotation about the origin to form a new linear transformation. The transformation formed sends the vector 34 to −3356.

Find the matrix representation of the transformation formed.

  • A 5 1 2 1 2 5
  • B 5 − 1 2 1 2 5
  • C − 1 1 0 0 1 4
  • D 5 − 1 2 1 2 5
  • E − 1 2 . 9 2 1 . 4 4 1 . 4 4 1 2 . 9 2

Find the scale factor of the original dilation.

  • Ascale factor = 169
  • Bscale factor = 13
  • Cscale factor = 154
  • Dscale factor = 13
  • Escale factor =√119

Q19:

The unit square, with vertices 𝑂(0,0),𝐴(1,0),𝐵(1,1), and 𝐶(0,1), is transformed by a rotation and then a dilation. Its image under this combined transformation is 𝑂𝐴𝐵𝐶, as shown in the diagram.

What are the coordinates of 𝐴′?

  • A 2 3 , 2 3
  • B 3 2 , 3 2
  • C 3 2 , − 3 2
  • D 2 3 , − 2 3
  • E 3 2 , 3 2

What is the matrix of the combined transformation?

  • A ⎡ ⎢ ⎢ ⎣ 3 2 − 3 2 − 3 2 3 3 ⎤ ⎥ ⎥ ⎦
  • B ⎡ ⎢ ⎢ ⎣ 3 2 3 2 − 3 2 3 2 ⎤ ⎥ ⎥ ⎦
  • C ⎡ ⎢ ⎢ ⎣ 2 3 − 2 3 2 3 2 3 ⎤ ⎥ ⎥ ⎦
  • D ⎡ ⎢ ⎢ ⎣ 3 2 − 3 2 3 2 3 2 ⎤ ⎥ ⎥ ⎦
  • E ⎡ ⎢ ⎢ ⎣ 2 3 − 2 3 − 2 3 2 3 ⎤ ⎥ ⎥ ⎦

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.